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This paper gives further insights about the Lyapunov-Krasovskiicharacterization of input-tostate stability (ISS) for switching retardedsystems on the basis of the results in [I. Haidar and P. Pepe.Lyapunov-krasovskii characterization of the input-to-state stability forswitching retarded systems. SIAM Journal on Control and Optimization,59(4):2997-3016, 2021]. We give new characterizations of the ISS propertythrough the existence of a relaxed common Lyapunov-Krasovskii functional. Moreprecisely, we show that the existence of a continuous Lyapunov-Krasovskiifunctional whose upper right-hand Dini derivative satisfies a dissipationinequality almost everywhere is necessary and sufficient for the ISS ofswitching retarded systems with measurable inputs and measurable switchingsignals. Different characterization results, using different derivativenotions, are also given.

ISS characterization of retarded switching systems with relaxed Lyapunov-Krasovskii functionals *